| 1 |
|
| 2 |
\section{\label{sec:DUFF}The DUFF Force Field} |
| 3 |
|
| 4 |
The DUFF (\underline{D}ipolar \underline{U}nified-atom |
| 5 |
\underline{F}orce \underline{F}ield) force field was developed to |
| 6 |
simulate lipid bilayer formation and equilibrium dynamics. We needed a |
| 7 |
model capable of forming bilayers, while still being sufficiently |
| 8 |
computationally efficient allowing simulations of large systems |
| 9 |
(\~100's of phospholipids, \~1000's of waters) for long times (\~10's |
| 10 |
of nanoseconds). |
| 11 |
|
| 12 |
With this goal in mind, we decided to eliminate all charged |
| 13 |
interactions within the force field. Charge distributions were |
| 14 |
replaced with dipolar entities, and charge neutral distributions were |
| 15 |
reduced to Lennard-Jones interaction sites. This simplification cuts |
| 16 |
the length scale of long range interactions from $\frac{1}{r}$ to |
| 17 |
$\frac{1}{r^3}$ (Eq.~\ref{eq:dipole} vs.~ Eq.~\ref{eq:coloumb}), |
| 18 |
allowing us to avoid the computationally expensive Ewald-Sum. Instead, |
| 19 |
we can use neighbor-lists and cutoff radii for the dipolar |
| 20 |
interactions. |
| 21 |
|
| 22 |
\begin{align} |
| 23 |
V^{\text{dipole}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 24 |
\boldsymbol{\Omega}_{j}) &= |
| 25 |
\frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 26 |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 27 |
- |
| 28 |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 29 |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 30 |
{r^{5}_{ij}} \biggr]\label{eq:dipole} \\ |
| 31 |
V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) &= \frac{q_{i}q_{j}}% |
| 32 |
{4\pi\epsilon_{0} r_{ij}} \label{eq:coloumb} |
| 33 |
\end{align} |
| 34 |
|
| 35 |
Applying this standard to the lipid model, we decided to represent the |
| 36 |
lipid model as a point dipole interaction site. Lipid head groups are |
| 37 |
typically zwitterionic in nature, with sometimes full integer charges |
| 38 |
separated by only 5 to 6~$\mbox{\AA}$. By placing a dipole of |
| 39 |
20.6~Debye at the head groups center of mass, our model mimics the |
| 40 |
dipole of DMPC.\cite{Cevc87} Then, to account for the steric hindrance |
| 41 |
of the head group, a Lennard-Jones interaction site is also located at |
| 42 |
the pseudoatom's center of mass. The model is illustrated in |
| 43 |
Fig.~\ref{fig:lipidModel}. |
| 44 |
|
| 45 |
\begin{figure} |
| 46 |
\includegraphics[angle=-90,width=\linewidth]{lipidModel.epsi} |
| 47 |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 48 |
is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
| 49 |
\label{fig:lipidModel} |
| 50 |
\end{figure} |
| 51 |
|
| 52 |
Turning to the tail chains of the phospholipids, we needed a set of |
| 53 |
scalable parameters to model the alkyl groups as Lennard-Jones |
| 54 |
interaction sites. For this, we used the TraPPE force field of |
| 55 |
Siepmann \emph{et al}.\cite{Siepmann1998} The force field is a |
| 56 |
unified-atom representation of n-alkanes. It is parametrized against |
| 57 |
phase equilibria using Gibbs Monte Carlo simulation techniques. One of |
| 58 |
the advantages of TraPPE is that is generalizes the types of atoms in |
| 59 |
an alkyl chain to keep the number of pseudoatoms to a minimum; the |
| 60 |
$\mbox{CH}_2$ in propane is the same as the central and offset |
| 61 |
$\mbox{CH}_2$'s in pentane, meaning the pseudoatom type does not |
| 62 |
change according to the atom's environment. |
| 63 |
|
| 64 |
Another advantage of using TraPPE is the constraining of all bonds to |
| 65 |
be of fixed length. Typically, bond vibrations are the motions in a |
| 66 |
molecular dynamic simulation. This necessitates a small time step |
| 67 |
between force evaluations be used to ensure adequate sampling of the |
| 68 |
bond potential. Failure to do so will result in loss of energy |
| 69 |
conservation within the microcanonical ensemble. By constraining this |
| 70 |
degree of freedom, time steps larger than were previously allowable |
| 71 |
are able to be used when integrating the equations of motion. |
| 72 |
|
| 73 |
After developing the model for the phospholipids, we needed a model |
| 74 |
for water that would complement our lipid. For this we turned to the |
| 75 |
soft sticky dipole (SSD) model of Ichiye \emph{et |
| 76 |
al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 77 |
in Sec.~\ref{sec:SSD}. The basic idea of the model is to reduce water |
| 78 |
to a single Lennard-Jones interaction site. The site also contains a |
| 79 |
dipole to mimic the partial charges on the hydrogens and the |
| 80 |
oxygen. However, what makes the SSD model unique is the inclusion of a |
| 81 |
tetrahedral short range potential to recover the hydrogen bonding of |
| 82 |
water, an important factor when modeling bilayers, as it has been |
| 83 |
shown that hydrogen bond network formation is a leading contribution |
| 84 |
to the entropic driving force towards lipid bilayer |
| 85 |
formation.\cite{Cevc87} |
| 86 |
|
| 87 |
BREAK |
| 88 |
|
| 89 |
END OF CURRENT REVISIONS |
| 90 |
|
| 91 |
BREAK |
| 92 |
|
| 93 |
|
| 94 |
|
| 95 |
|
| 96 |
|
| 97 |
The main energy function in OOPSE is DUFF (the Dipolar Unified-atom |
| 98 |
Force Field). DUFF is a collection of parameters taken from Seipmann |
| 99 |
and The total energy of interaction is given by |
| 100 |
Eq.~\ref{eq:totalPotential}: |
| 101 |
\begin{equation} |
| 102 |
V_{\text{Total}} = |
| 103 |
\overbrace{V_{\theta} + V_{\phi}}^{\text{bonded}} + |
| 104 |
\underbrace{V_{\text{LJ}} + V_{\text{Dp}} + % |
| 105 |
V_{\text{SSD}}}_{\text{non-bonded}} \label{eq:totalPotential} |
| 106 |
\end{equation} |
| 107 |
|
| 108 |
\subsection{Bonded Interactions} |
| 109 |
\label{subSec:bondedInteractions} |
| 110 |
|
| 111 |
The bonded interactions in the DUFF functional set are limited to the |
| 112 |
bend potential and the torsional potential. Bond potentials are not |
| 113 |
calculated, instead all bond lengths are fixed to allow for large time |
| 114 |
steps to be taken between force evaluations. |
| 115 |
|
| 116 |
The bend functional is of the form: |
| 117 |
\begin{equation} |
| 118 |
V_{\theta} = \sum k_{\theta}( \theta - \theta_0 )^2 \label{eq:bendPot} |
| 119 |
\end{equation} |
| 120 |
$k_{\theta}$, the force constant, and $\theta_0$, the equilibrium bend |
| 121 |
angle, were taken from the TraPPE forcefield of Siepmann. |
| 122 |
|
| 123 |
The torsion functional has the form: |
| 124 |
\begin{equation} |
| 125 |
V_{\phi} = \sum ( k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0) |
| 126 |
\label{eq:torsionPot} |
| 127 |
\end{equation} |
| 128 |
Here, the authors decided to use a potential in terms of a power |
| 129 |
expansion in $\cos \phi$ rather than the typical expansion in |
| 130 |
$\phi$. This prevents the need for repeated trigonometric |
| 131 |
evaluations. Again, all $k_n$ constants were based on those in TraPPE. |
| 132 |
|
| 133 |
\subsection{Non-Bonded Interactions} |
| 134 |
\label{subSec:nonBondedInteractions} |
| 135 |
|
| 136 |
\begin{equation} |
| 137 |
V_{\text{LJ}} = \text{internal + external} |
| 138 |
\end{equation} |
| 139 |
|
| 140 |
|
